
where the respective last step holds under symmetric integration.
Finally, using Fierz identities we can express \((\hat{O}^{LL} )_\alpha ^\alpha \)  and
\((\hat{O}^{LR} )_\alpha ^\alpha \)  in terms of the four-quark scalar operator basis used in
Ref. [1]}, obtaining
\(g^{\mu \nu } \, \hat{\Pi }_{\mu \nu }^{LL} (k,0) &= \frac{ i g_s^2}{(k^2 + i \epsilon )^2} \ 2 O_1 \,,\\g^{\mu \nu } \, \hat{\Pi }_{\mu \nu }^{LR} (k,0) &= \frac{ i g_s^2}{(k^2 + i \epsilon )^2} \ 2 \bar{O}_{LR}\,,\qquad \bar{O}_{LR} = \frac{1}{2} \left( O_4 - 3 O_5 \right) \,.\) 
